The thermo-viscous fingering instability of a cooling spreading liquid dome

Abstract

We investigate a theoretical model of a molten viscous planar liquid dome spreading under gravity over an inclined substrate. The liquid in the dome cools as it spreads, losing its heat to the surrounding colder air and substrate. Coupled nonlinear evolution equations for the dome's thickness and temperature describing the spreading flow are derived employing the lubrication approximation. The coupling between the flow and cooling is via a temperature-dependent viscosity. For intermediate Péclet numbers, a new one-dimensional free surface shape is identified. In this solution, the hotter and more mobile liquid piles up behind the dome's colder and less mobile leading edge, forming a distinct elevated ridge at the flow front. The ridge solution is mapped in parameter space. The transverse stability of the one-dimensional ridge solution is investigated using linear stability analysis and numerical simulations. The existence of a thermo-viscous fingering instability is revealed. For this instability to occur, the presence of the ridge is shown to be necessary. Two-dimensional simulations confirm the stability analysis elucidating the underlying thermo-viscous mechanism.